Point Group Tables

Point Group Tables

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Point Group Tables of D₃(32)

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Character Table of the group D₃(32)*
D₃(32)	#	1	3	2_1-10	functions
Mult.	-	1	2	3	·
A₁	Γ₁	1	1	1	x²+y²,z²
A₂	Γ₂	1	1	-1	z,J_z
E	Γ₃	2	-1	0	(x,y),(xz,yz),(x²-y²,xy),(J_x,J_y)

**Subgroups of the group D₃(32)**
Subgroup	Order	Index
D₃(32)	6	1
C₃(3)	3	2
C₂(2)	2	3
C₁(1)	1	6

[ Subduction tables ]

**Multiplication Table of irreducible representations of the group D₃(32)**
D₃(32)	A₁	A₂	E
A₁	A₁	A₂	E
A₂	·	A₁	E
E	·	·	A₁+A₂+E

[ Note: the table is symmetric ]

**Symmetrized Products of Irreps**
D₃(32)	A₁	A₂	E
[A₁ x A₁]	1	·	·
[A₂ x A₂]	1	·	·
[E x E]	1	·	1

**Antisymmetrized Products of Irreps**
D₃(32)	A₁	A₂	E
{A₁ x A₁}	·	·	·
{A₂ x A₂}	·	·	·
{E x E}	·	1	·

**Irreps Decompositions**
D₃(32)	A₁	A₂	E
V	·	1	1
[V²]	2	·	2
[V³]	1	3	3
[V⁴]	4	1	5
A	·	1	1
[A²]	2	·	2
[A³]	1	3	3
[A⁴]	4	1	5
[V²]xV	2	4	6
[[V²]²]	6	1	7
{V²}	·	1	1
{A²}	·	1	1
{[V²]²}	2	3	5

V ≡ the vector representation
A ≡ the axial representation

**IR Selection Rules**
IR	A₁	A₂	E
A₁	·	x	x
A₂	x	·	x
E	x	x	x

[ Note: x means allowed ]

**Raman Selection Rules**
Raman	A₁	A₂	E
A₁	x	·	x
A₂	·	x	x
E	x	x	x

[ Note: x means allowed ]

**Subduction of the rotation group D(L) to irreps of the group D₃(32)**
Irreps	Dimensions	Irreps of the point group
L	2L+1	A₁	A₂	E
0	1	1	·	·
1	3	·	1	1
2	5	1	·	2
3	7	1	2	2
4	9	2	1	3
5	11	1	2	4
6	13	3	2	4
7	15	2	3	5
8	17	3	2	6
9	19	3	4	6
10	21	4	3	7

^* C. J. Bradley and A. P. Cracknell (1972) The Mathematical Theory of Symmetry in Solids Clarendon Press - Oxford
^* Simon L. Altmann and Peter Herzig (1994). Point-Group Theory Tables. Oxford Science Publications.

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Point Group Tables of D3(32)

Point Group Tables of D₃(32)